Tuesday, December 28, 2010

"These allusions give incidentally some indication of the disproportionate length of time over which so short a work has been in preparation... As was only natural during such a time period, others have from time to time independently taken up points of view which are similar to one or the other adopted in this paper and have developed them further or in different directions from those pursued here." -- Piero Sraffa

Despite my appreciation of Bliss' 1975 book, I think the following view dubious, uninformed, and authoritarian:

"A striking feature of the school to which Piero Garegnani belongs is its seeming lack of interest in the real world... Our world is changing rapidly and in ways that demand economic analysis of what is happening... Over the last 30 years so-called neoclassical economics has been extraordinarily productive... What has been the contribution of the post-Sraffa school in the same period? Nothing at all as far as I can see. This has been an exceptionally sterile approach. Where are the new ideas? Where are the illuminating insights into what is happening today?" -- Christopher Bliss (2009)

I have no problem with a researcher deciding to center their investigations into criticism and the history of economic thought. I think that when Bliss calls neoclassical economics "extraordinarily productive", he includes research that fails to test neoclassical economics and whose relationship to neoclassical economics can be doubted. Sraffian economics can easily exceed this standard.

I take the "others" Sraffa refers to above to be principally Wassily Leontief and John Von Neumann. So, for example, work with Leontief's Input-Output (I/O) analysis is applied Sraffian analysis. Interestingly enough, countries maintain their national accounts in a form supporting I/O analysis. (For the United States, see the benchmark input-output accounts available from the Bureau of Economic Analysis (BEA). For many developed countries, see the Structural Analysis (STAN) Database for data on Industry and Services available from the Organisation for Economic Co-Operation and Development (OECD).) For me, challenges in working with this data arise from statistical discrepancies, rectangular matrices that I expect to be square, components in value added that are neither wages nor profits, import and export flows, etc. But other economists, including some Sraffians have addressed these challenges in their own work.

I have listed selected applied Sraffa work on twotopics.. Tony Aspromourgos (2004) lists Sraffian research applied to a larger range of issues.

Wednesday, December 22, 2010

Figure 1: From Sraffa's Production of Commodities by Means of Commodities

Suppose people save more. A neoclassical and Austrian school1 idea is that then the supply of capital will have increased, in some sense. One would expect its price, the rate of interest, to be less, absent intervention by the monetary authorities. Entrepreneurs, if they were alert, would adopt more capital-intensive - that is - longer techniques of production. After these techniques came online, output per worker would be higher.

Suppose that the length of the period of production of a technique were defined in terms of purely physical data. Given a complete list of inputs and outputs, including the times at which they flow into and out of the production processes, one would be able to measure the (average) period of production of the technique composed of those processes. Then reswitching shows the above story cannot be universally valid.

Austrian-school economists and economists sympathetic to the Austrian school have had at least two reactions to this demonstration of the logical invalidity of the above story. One reaction is to assert that an aggregate measure of capital-intensity, such as a physical measure of the average period of production, is not needed for the story to go through. Supposedly, entrepreneurs shift resources, in response to low interest rates, to time periods further before the production of consumer goods. In the technical jargon, entrepreneurs tend to move resources towards producing goods of higher orders and away from producing goods of lower orders. I have shown2 that this response fails to evade a critique based on reswitching.

The second reaction is to define the average period of production as dependent on the interest rate, as well as physical properties of techniques of production. Thus, the average period of production for a given technique will be different at the interest rates associated with different switch points. Apparently, Edmond Malinvaud, drawing on some work of J. R. Hicks, made this argument in a 2003 paper about Knut Wicksell's contributions. Saverio Fratini, in his paper presented at the recent 50th anniversary conference on Sraffa's book, took the opportunity of Malinvaud's paper to argue that this reaction also cannot be sustained as a response to a critique based on reswitching and capital-reversing.

I associate this second reaction with Leland Yeager, who, in a couple of papers in the late 1970s, argued that waiting has the dimensions of the product of value and time (for example, dollar-years). I find it hard to find a valid non-tautological argument here. I think Fratini's paper could be revised to point out that he refutes Yeager, as well as J. R. Hicks and Edmond Malinvaud. I would like to be referenced too, although I don't know about the conventions of referencing working papers.

Footnotes1 Both William Stanley Jevons and Eugen von Böhm-Bawerk expounded this idea.2 Due to a recent hard-disk crash, I have lost originals from which I generated some PDFs, as well as reviewer comments on recent revisions.References

Monday, December 20, 2010

In comments on my previous post, a blogger from Revista Circus links to a presentation of Gary Mongiovi on Marxian exploitation. This is too modest. The blog has organized a plethora of videos, abstracts, and draft papers from the recently completed international conference in Rome on Sraffa's Production of Commodities by Means of Commodities:

Saturday, December 11, 2010

Michele Boldrin has written a paper in which supposedly Marxian themes are treated in a Dynamic Stochastic Equilibrium Model (DSGE). He writes:

...there is 'exploitation of labor' also in standard neoclassical models of production. Within each firm, almost all workers (i.e. everyone but the marginal worker at the marginal plant) are 'exploited' in the sense that they are being paid a wage smaller than their marginal productivity. -- Michel Boldrin (2009) "Growth and Cycles, in the Mode of Marx and Schumpeter. Scottish Journal of Political Economy, V. 56, N. 4 (p. 432)

The above is mistaken on at least two points:

The notion of exploitation is Joan Robinson's neoclassical idea from the period in which she developed the theory of imperfect competition; this idea is not Marx's.

The above passage seems to take the value of the marginal product of labor as defined prior to prices, including wages. If so, Boldrin follows the mathematically mistaken teaching of some not-so-bright orthodox economists.

Boldrin continues his mistaken line:

From a Marxian view point, labor-saving innovations are the means through which capitalist exploitation can be perpetrated and maintained over time... -- Michel Boldrin (2009)(p. 435)

The above might or might be true of Marxian exploitation, but Boldrin is using a different definition. And again:

Asymptotically, all existing firms use the same, best, technology and the market wage corresponds to the marginal productivity of labor in the marginal technology, which is also the one everybody uses. Hence, exploitation of the workers has ceased. -- Michel Boldrin (2009)(p. 440)

John Roemer describes a source of profits in a model which could exhibit perfect free entry, constant returns to scale, and individual profit maximization:

"...the existence of postive-profit equilibria ... is to associated with the necessity of time in production, that capitalists must advance the costs of production before they receive the revenues from production. It is this temporal structure of production that gives rise to the economic necessity of a capital constraint, whether or not funds for production are limited to internal finance or are available on a capital market." -- John E. Roemer (1981). Analytical Foundations of Marxian Economic Theory, Cambridge University Press (p. 84)

As those familiar with Frank Hahn's critique of the "neo-Ricardians" know, this sort of model is consistent with every valid marginal productivity principle holding.

Man-Seop Park's criticizes new growth theory (from, for example, Paul Romer) on a number of grounds in a number of papers. One of these grounds is that such models ignore the presence of time in production, even when they depict a number of stages in production. I think Boldrin's paper may be weak on this ground.

Update: I thought a little more about this. Bouldrin considers the case in which the marginal plants in both the investment and consumer goods sector are both operated at less than capacity. In this case, the value of the marginal product of labor is positive (in the competitive case), and the marginal product of the capital good is zero.

I would rather consider the case in which both labor and the marginal plants are binding in both sectors. In this case, an additional unit of either labor or plant would contribute nothing to production. On the other hand, a marginal unit decrease in either labor or capital would decrease production by some specified amount. Thus, the value of the marginal product of both labor and the capital good (in the competitive case) would be an interval from zero to some positive amount. I think this case results in the familiar Sraffian wage-rate of profits curve in which wages can only be larger if the rate of profits would be smaller.

Wednesday, December 08, 2010

Here Wolff provides an overview of Marx, agrees with Morishima that Marx was a great economist, and mentions books by the analytical Marxists. Here he describes Sraffa's impact on interpretations of Marx and provides a list of books (which overlaps with my list of textbooks). Here he briefly overviews each of the books in his list, with the exception of Marglin's.

Saturday, December 04, 2010

David Ruccio hypothesizes that the current worldwide macroeconomic difficulties are related to increased inequality in the distribution of income in, say, the United States over the last few decades. I have been considering hypothetical mechanisms that expand on this idea. Previously I have put forth the Harrod-Domar growth model as a framework in which increased inequality leads to a tendency towards recessions. In this post, I focus on causation in the other direction. (A full analysis of the issues will likely describe a process of cumulative causation.)

Active macroeconomic fiscal policy is assisted if government spending is already a somewhat large component of a nation's economy. It is easier to raise government spending by some given fraction of national income if that change is a smaller percentage of current government spending. We have seen this issue in connection with the Obama administrations talk of "shovel-ready" projects and the stimulus policy. Perhaps this consideration even applies to automatic stabilizers.

So governments that are smaller with respect to their national economies might be expected to exhibit worse macroeconomic performance. And James Galbraith has shown quite some time ago that poor macroeconomic performance leads to increased greater inequality in wages.

Perhaps the above is part of the explanation for the empirical cross-section correlation between decreased government size and increased inequality. I think Hacker and Pierson give some explanation why increased inequality engenders political forces tending towards smaller government size.

Saturday, November 27, 2010

Joan Robinson, drawing on John Maynard Keynes, famously distinguished between models set in historical and logical time. Geoffrey Hodgson, now an institutionalist economist, has written extensively about evolutionary models in economics. Given my interest in such economists, I also find of interest how to formalize the notion that "history matters". I think mathematicalmodels of nonergodic processes is one way of formally setting a model in historical time.

Brian Arthur and Paul David, two economists, have developed a parallel idea, that of path dependence. This post is about false statements Stan Liebowitz and Stephen Margolis like to make about this work. Liebowitz and Margolis quote Paul David:

"The foregoing account of what the term 'path dependence' means may now be compared with the rather different ways in which it has come to be explicitly and implicitly defined in some parts of the economics literature. For the moment we may put aside all of the many instances in which the phrases 'history matters' and 'path dependence' are simply interchanged, so that some loose and general connotations are suggested without actually defining either term. Unfortunately much of the non-technical literature seems bent upon avoiding explicit definitions, resorting either to analogies, or to the description of a syndrome - the set of phenomena with whose occurrences the writers associate path dependence. [Rather than telling you what path dependence is, they tell you some of the symptomology - things that may, or must happen when the condition is present. It is rather like saying that the common cold is sneezing, watering eyes and a runny nose.]" -- Paul David

Liebowitz and Margolis somehow think you will be persuaded to believe the following:

"So here we see David disqualifying, at least from others, any efforts to connect path dependence to observable phenomena. David would have path dependence discussed only in the context of the most severe abstraction, an immaculate concept immune from criticism: it is a dynamic stochastic process that is non-ergodic." -- Stan Liebowitz and Stephen Margolis

Notice Paul David never says that path dependence, under a rigorous definition, never will be manifested in observable empirical phenomena. Elsewhere Paul David notes that Markov processes can be non-ergodic, that is, path dependent. And he notes that economists have connect Markov processes, not all of which need be path-dependent, to observable penomena:

"Homogeneous Markov chains are familiar constructs in economic models of the evolving distribution of workers among employment states, firms among size categories, family lineages among wealth-classes or socio-economic (occupational) strata, and the rankings of whole economies among in the international distribution of per capita income levels." -- Paul David

Sunday, November 14, 2010

The appendix of Omkarnath's aprreciation of Bharadwaj's review of Sraffa's book contains an exchange of letters between Bharadwaj and Sraffa. Here's Sraffa's letter:

"As from Trinity College, CambridgeSabzburg, AustriaSeptember 8, 1963

Dear Mrs. Bharadwaj,

Thank you so much for your kind letter, which I receive abroad, just as I am about to return to Cambridge.

I am delighted with your excellent article, which will be of great help to many who have been puzzled by my book. I have no doubt that you have correctly grasped the main lines of the argument and also guessed some of the directions in which, in my view the criticism of marginalism should be developed.

There are only one or two minor points in which I think your exposition is perhaps not as lucid as it is in general. One (p. 1452, col 3, first para), on deriving 'land theory of value'. I suppose this is an echo of Samuelson, but I do not see the implications. (I have the impression that you somewhat underestimate the historical importance of the labour theory of value: in my view much of modern theory is, whether consciously or not, a polemic against it.) Another one is p. 1453, col 1, from 'To a particular wage' to the end of the paragraph: I find this obscure and have some doubts about what I seem to understand. Also, a small point of disagreement is the very last words of your final footnote. I insist that the construction of a subsystem is a purely bookkeeping operation!

These, however, are very minor blemishes and the article remains a splendid contribution for which I am most grateful. I hope very much you will continue to work on these lines.

With kind regards,

Yours sincerely,Piero Sraffa"

Sraffa criticizes three passages in Bharadwaj's review. I agree the second passage is not well written:

"To a particular wage, given in any standard, there may correspond several alternative rates of profits. This shows the absolute necessity of measuring the wage in terms of the Standard commodity, if unequivocal conclusions regarding the movements of the rate of profits given the wage rate are to be drawn. Measurement of the wage in terms of the Standard commodity gives us definite information regarding both the direction as well as the extent of change in the rate of profits, consequent upon a change in the wage rate. No other standard possesses this predictive value." -- Krishna Bharadwaj (1963)

The wage is only defined in terms of some standard. Once the standard is specified, a unique rate of profits corresponds to that wage, no matter what the standard. A different rate of profits may correspond to the same numeric value in a different standard. But one would not expect the same properties to be displayed by a substance at zero degrees Celsius and zero degrees Fahrenheit.

I think Sraffa's criticism of the third passage is a disagreement about the level of emphasis. Here is that passage:

"Nevertheless, while reading the paragraphs relating to the construction of the Standard system (pp. 23-4) and more particularly the subsystems (p. 89), one gets a feeling as though the assumption of constant returns-to-scale is necessary. Such doubts could easily be warded off since the Standard commodity is purely an auxiliary construction having no physical existence in production relations. Similar is the case of the subsystems which are used to derive the direct and indirect labour content of commodities (at zero profit rate)." -- Krishna Bharadwaj (1963)

Tuesday, November 09, 2010

Some comic gives some stupid and vile economists an opportunity to comment on some of Cosma Shalizi's ideas [1]. (I find particularly stupid the commentator that cites Amazon, a Web 2.0 exemplar, for an example of agents interacting only through prices.) Eventually, one respondent says that George-Marios Angeletos and Jennifer La'O show how to embed animal spirits in Dynamic Stochastic General Equilibrium (DSGE) models. I'm not sure which paper they are talking about, but I did download their paper Sentiments (with accompanying slides.) I do not object to how Angeletos and La'O model shocks, although in this setting I don’t see why I should care.

I was quickly stopped in reading this paper at the first footnote:

"By 'mainstream' we mean the prototypical RBC and New-Keynesian models, as well as the more recent DSGE models. This excludes models with multiple equilibria or irrational agents, which we discuss in due course."

How are models with multiple equilibria non-mainstream? They continue in this vein throughout their paper:

"In our model, agents are fully rational; preferences and technologies are standard; markets are Walrasian; there are no nominal frictions, no externalities, and no non-convexities; the equilibrium is unique; and there is no room for correlation devices or lotteries. In these respects, our theory has squarely neoclassical foundations."

"This extrinsic uncertainty has very similar flavor as the one encountered in models with multiple equilibria: it captures the self-fulfilling nature of short-run fluctuations. Importantly, though, it does not rest on the severe externalities, non-convexities and missing markets that are most often needed to sustain multiple equilibria-nor does it come with the usual difficulties in conducting policy analysis."

It is my understanding that the Sonnenschein-Mantel-Debreu theorem is proved in a Walrasian model with no externalities, non-convexities, or missing markets. The excitement over the theorem comes from being derived with the same sort of assumptions that are used in Debreu's Theory of Value to derive the existence of an equilibrium. So I find it hard to believe the authors know what they are talking about when they suggest multiple equilibria are non-Walrasian or non-neoclassical.

By the way, I happen to have available a model with multiple equilibria. Figures 1 and 2 below illustrate. I'd like somebody to point out to me how agents in this model are not fully rational; how preferences and technologies are non-standard; or where nominal frictions, externalities, and non-convexities exist in this model. I suppose one can say some markets do not exist in an overlapping generations model. Agents cannot buy commodities or sell their labor before they are born or after they are dead. I did not think such an assumption made a model heterodox or non-neoclassical.

Figure 1: Equilibrium Interest Rates as a Function of One Parameter in the Utility Function

Figure 2: Equilibrium Wages as a Function of Another Parameter

[1] Cosma Shalizi is on a team of three that has recently received a grant awarded by the Institute for New Economic Thinking.

Friday, November 05, 2010

Has the global financial crisis discredited Dynamic Stochastic General Equilibrium (DSGE) models? It seems to me that that may be so, but I wonder if new criticisms of DSGE models have been put forth. It seems to me the fundamental flaws of such models have been unaddressed for decades. Cosma Shalizi describes these models in a critical way. Joseph Stiglitz is not impressed with them either:

"It is hard for non-economists to understand how peculiar the predominant macroeconomic models were. Many assumed demand had to equal supply - and that meant there could be no unemployment. (Right now a lot of people are just enjoying an extra dose of leisure; why they are unhappy is a matter for psychiatry, not economics.) Many used "representative agent models" - all individuals were assumed to be identical, and this meant there could be no meaningful financial markets (who would be lending money to whom?). Information asymmetries, the cornerstone of modern economics, also had no place: they could arise only if individuals suffered from acute schizophrenia, an assumption incompatible with another of the favoured assumptions, full rationality." -- Joseph Stiglitz

As I understand it, Smets and Wouters (2007) is an example of a DSGE model widely approved of by mainstream economists. Sbordone et al. (2010) is a recent presentation of an introductory DSGE. One can see that the output of these models is a set of stochastic processes meant to model certain time series available in empirical data. Nominal interest rates, (real) income, the inflation rate, the volume of one-period government bonds, employment, and nominal wages are all examples of such time series. The input into such models is another set of stochastic processes. These inputs are given names that suggest they are random terms in functions characterizing either government entities - e.g., monetary policy shock - or agents in microeconomic models. Examples of the latter kind of names are a household discount rate shock, productivity shock, markup shock, and firm discount rate shock. Stochastic processes are specified by parameters of certain probability distributions. As one can see from the names of these inputs, the agents are supposed to be optimizing, including across time. A story, expressed in mathematics and supposedly of microeconomic equilibrium, connects the inputs to the outputs in the model. That is, the DSGE models are supposed to have microfoundations.

But they do not have microfoundations. I look for a number of mistakes in such models:

Are inputs into production function measured in numeraire units? (The numeraire is often taken to be a basket of consumer goods.) Joan Robinson (1953-54) explains why measuring the quantity of capital in production functions in numeraire units is an error. Notice this is not solely a question of the aggregation of capital. A model can have a continuum of capital goods, yet still exhibit this mistake.

Are representative agents used? Kirman (1992) explains why the use of representative agents is unfounded.

Is money modeled? Frank Hahn (1965) explains why money does not matter in General Equilibrium models, even though it does seem to matter for actually existing capitalist economies. Mainstream economists have a couple of strategies for introducing money in an ad hoc way into DSGE models. But I am not convinced the typical modeler has ever managed to address Hahn's point.

Is the possibility of multiple equilibria taken seriously? Is it demonstrated that non-equilibrium dynamic processes converge to the modeled equilibrium? Richard Goodwin (1990) illustrates what a macroeconomics looks like that, in contrast to typical DSGE models, takes dynamics seriously. Kirman (1989) shows that ignoring muliple equilibria and stability issues was demonstrated to be unfounded by the Sonnenschein-Mantel-Debreu results. Shiller (1978) long ago raised the issues of multiple equilibria and convergence. Shiller was critiquing the tradition out of which DSGE models evolved.

I haven't read much in the literature of DSGE models. It seems to me, however, that these issues are routinely ignored by many mainstream economists. Perhaps a wider range of macroeconomic models should be considered by serious researchers.

Tuesday, November 02, 2010

I think that if one looked, one would be able to find in lots of depictions in literature of multiple selves. Here's an example:

"...he had heard about him the constant voices of his father and of his masters, urging him to be a gentleman above all things and urging him to be a good catholic above all things. These voices had now come to be hollow-sounding in his ears. When the gymnasium had been opened he had heard another voice urging him to be strong and manly and healthy and when the movement towards national revival had begun to be felt in the college yet another voice had bidden him to be true to his country and help to raise up her language and tradition. In the profane world, as he foresaw, a worldly voice would bid him raise up his father's fallen state by his labours and, meanwhile, the voice of his school comrades urged him to be a decent fellow, to shield others from blame or to beg them off and to do his best to get free days for the school. And it was the din of all these hollow-sounding voices that made him halt irresolutely in the pursuit of phantoms." -- James Joyce, A Portrait of the Artist as a Young Man

Does how artists depict human beings carry any weight for how economists choose to portray agent's choices? Should it?

Saturday, October 30, 2010

Every once in a while, I notice popular and semi-popular journals in which ideas I find interesting are discussed. As I understand it, Economic & Political Weekly is an Indian journal. It was named Economic Weekly when it published KrishnaBharadwaj's 1963 review of Sraffa's book. I cannot speak to Indian politics. But I find two articles of interest in the 16 October number.

Ghose (2010) builds on Arthur Lewis's model of development. Lewis depicts undeveloped economies as exhibiting a kind of dualism in which capitalist and traditional (subsistence) sectors coexist. The traditional sector experiences disguised underemployment and can provide an infinite supply at labor at the going wage. Lewis mentions farmerss, casual workers, petty traders, domestic and commercial retainers, and woman in the household as sources of such a labor supply. Economic development is a matter of structural change in which the capitalist sector expands and replaces the traditional sector. My impression is that this distinction between structural change and a mere quantitative expansion used to distinguish the economics of development and of growth. Ghose suggests this perspective has been lost in development economics.

Sinha (2010) provides an appreciation of Sraffa's book. Sinha does not see Sraffa's prices of production as dynamically stable limit points of some sort of gravitational process governing market prices. Nor does he think an internal critique of neoclassical economics based on reswitching was central to Sraffa's project. He emphasizes Sraffa's standard commodity and likes Joan Robinson's reading of Sraffa as generalizing Ricardo's corn economy, in which the rate of profits is a physically specified ratio independent of relative prices. This reading resembles Bharadwaj's.

Tuesday, October 26, 2010

Brad DeLong has kindly made his lecture notes on General Equilibrium available online. I think he explicitly only asserts that equilibria exist (under certain conditions), not that any are necessarily stable. But I do not see how any student reading his notes cannot come to that conclusion. He never explicitly states that economists have found that General Equilibrium Theory imposes basically no limit on dynamics - this is an implication of the Sonnenschein-Mantel-Debreu results. I also don't care for DeLong's treatment here of Karl Marx, who was not writing about the allocation of given resources. (I think that a formalization of Marx's notion of prices of production is more like a Von Neumann ray without requiring labor markets to clear.)

Not being a teacher, I'm willing to entertain discussion of simplifications in teaching beginners. I can see why Joan Robinson thought that General Equilibrium Theory doesn't stand up long enough to be knocked down. The complete model postulates that enough markets exist such that you can trade any commodity for any other commodity across all time periods and all states of nature. This is wildly non-descriptive of any actual capitalist economies. But I can see how the introductory teacher might not get to point where this objection makes any sense.

Sunday, October 24, 2010

Peter Nobel, who is descended from Alfred Nobel, has come out against the Nobel Prize in economics. I want to focus on this part of his statement:

"With no knowledge of economics, I have no opinions about the individual economics prize winners. But something must be wrong when all economics prizes except two were given to Western economists, whose research and conclusions are based on the course of events there, and under their influence."

My question is which two prize winners does Nobel have in mind? I think Leonid Kantorovich, the Soviet co-inventer of Linear Programming, is obviously one. Is the other Amartya Sen? I think one might contrast the West with both Eastern block countries and the global South. If one counts the South as non-West, then Peter Nobel's count is not quite correct. One would also have to count Arthur Lewis, for his contributions to development economics.

Thursday, October 21, 2010

I seem to be very slow to either read these or write up a detailed explanation:

Francis M. Bator (1958) "The Anatomy of Market Failure", Quarterly Journal of Economics, V. 72, N. 3 (Aug): 351-379. John Cassidy takes this paper as the authoritative definition in his book How Markets Fail: The Logic of Economic Calamities.

Arindrajit Dube, T. William Lester, and Michael Reich (2008) "Minimum Wage Effects Across State Borders: Estimates Using Contiguous Counties", forthcoming in the Review of Economics and Statistics. Generalizes the natural experiment approach of Card and Krueger to look at all cross-state local differences in minimum wages in the United States between 1990 and 2006. They find no adverse employment effects from higher minimum wages in the ranges examined. You can watch a video interview with Dube here. (The Wikipedia page on minimum wages also lists meta-analyses, by Stanley and by Doucouliagos & Stanley, more recent than Card and Krueger's meta-analysis.)

Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou (2009) "The Complexity of Computing a Nash Equilibrium", Communications of the ACM, V. 52, No. 2: pp. 89-97. Defines a complexity class between P and NP and proves that computing a Nash equilibrium is in that class. Thus, if P ≠ NP, Nash equilibria cannot be computed in polynomial time for arbitrary games. In other words, computing a Nash equilibrium in general is infeasible in practice. (Tim Roughgarden's "Algorithmic Game Theory" (Communications of the ACM, V. 53, No. 7 (Jul. 2010)) and Yoav Shoham's "Computer Science and Game Theory" (Communications of the ACM, V. 51, No. 5 (Aug. 2008)) are survey articles.)

P.S. Commentator Emil Bakhum lists some objections to Sraffa's analysis from Alfred Muller. I do not agree that these objections correctly characterize Sraffa's analysis, a point to which I may return. I think I would like a more complete reference, although I might not be able to read it if it is in german.

Saturday, October 16, 2010

Peter Diamond, Dale Mortensen, and Christopher Pissarides won the "Nobel" prize in economics this year. I do not have Bill Mitchell’s expertise on search theory and labor economics. Nevertheless, I thought I’d record my reactions.

As I understand it, Diamond, Mortensen, and Pissarides think labor would be described by by the interactions of well-behaved supply and demand functions for labor if it were not for the heterogeneity of workers and jobs and the time to form matches between them. The orthodox theory would be wrong even if workers and jobs were homogeneous. So I find puzzling why I should approve of this year’s award. I think my opinion is consistent with some of Alessandro Roncaglia’s observations of trends in mainstream economics.

Tuesday, October 12, 2010

I guess this is part of aseries in which I describe oddities upon which I have stumbled. Here I focus on two phenomena in the measurement of gravity. Perhaps the theory of general relativity is wrong. (The link goes to an explanation of a different problem in physics). Of course, much more prosaic explanations are possible.

Maurice Allais, the recently dead "Nobel" laureate in economics, experimented with a paraconical pendulum during the 1950s. He discovered the Allais effect, which is a variation in the behavior of a pendulum during an eclipse. The plane of the pendulum rotated approximately ten degrees during the eclipse and then returned to the previously pattern. A number of scientists have tried to replicate this and similar effects with various experimental equipment during various eclipses. Some succeeded in replication and some failed. Allais’ explanation, apparently, was to revive the 19th century concept of the aether and argue that space is anisotropic.

Pioneers 10 and 11 were launched in 1972 and 1973, respectively. NASA was still in communication with them after they had passed beyond the orbit of Pluto and were more than 20 Astronomical Units away from the sun. (An AU is the average distance from the Earth to the sun.) Pioneers 10 and 11 have an anomalous acceleration towards the sun of an order of magnitude of 10-7 centimeters per square seconds. In other words, as they move away from the sun, they are very slowly slowing down more than can be accounted for under the current (relativistic) understanding of gravity. Pioneers 10 and 11 are moving away from the sun at a rate of approximately 12 kilometers per second. (It dawned on me while writing the above that I am no longer sure how many planets are in our solar system.)

References

Maurice Allais (1986). "The Concepts of Surplus and Loss and the Reformulation of the Theories of Stable General Equilibrium and Maximum Efficiency", in Foundations of Economics (edited by Mauro Baranzini and Roberto Scazzieri), Basil Blackwell.

Friday, October 08, 2010

Stephen Williamson writes some ignorant balderdash about heterodox economics. At least two of his commentators recognize the quality of his remarks.

If Williamson had a clue, he would know some scholars distinguish between the heterodox/orthodox distinction and the mainstream/nonmainstream distinction. Following Davis's taxonomy, I would classify North and new institutionalist economics as "mainstream heterodox". I do not think of either Paul Krugman or James Tobin as non-mainstream.

Non-mainstream heterodox economics do not all reject or dislike mathematics. Williamson doesn't seem to know about the existence of sraffians. Of course, he doesn't list feminist economics as a target of his ill-informed calumny. I think most heterodox economists would agree that one can make true and insightful statements about economics without using mathematics. One capable of a moment's reflection can see that that doesn't imply no mathematics can be be useful for economics. Williamson seems to think that the point of using mathematics is to seperate oneself from the hoi polloi. It is no legimate criticism of a paper that a graduate student can write out the model in twenty minutes. The questions one should ask oneself, in this case, are "Does this paper provide empirical and policy guidance for understanding some aspect of Japan's economy?" and "Is this paper original?"

Consider such journals as the American Economics Review or the Journal of Political Economy. And consider the Cambridge Journal of Economics, Metroeconomica, or the Review of Political Economy. The first set contains examples of mainstream economics. The second set of journals publish non-mainstream heterodox economists. An outsider will find journals in both set contain a mixture of natural language and mathematics. They both have both theoretical and empirical papers. The academics are not all from one university, they argue with one another, and they seem to have a variety of sources of funding. Clearly, non-mainstream heterodox economists are not "fringe" in the same sense that most are who argue for astrology, creationism, or a flat earth theory.

Saturday, October 02, 2010

National Public Radio broadcast an interview last evening with Charles Ferguson, the director of the soon-to-be-released documentary, Inside Job. It is apparently about recent financial shenanigans on Wall Street and the proximate causes of the global financial crisis. The director seemed to be struck about how unlikely it is that top financiers would be heading off to prison. He also was astonished about the pervasive advocacy and lack of ethics he found among economists. He mentioned economists writing articles to promote some industry and not disclosing funding, serving on corporate boards, and acting as expert witnesses in court cases.

NPR played a clip where Glenn Hubbard, the dean of the Columbia University School of Business, a former chair of the Council of Economic Advisors, and a Visiting Scholar (if you can call it that) at the American Enterprise Institute, basically threw Ferguson out of his office. Hubbard apparently did not want to talk about his outside sources of income.

Update: In the comments, Tomboktu links to the NPR piece. Elsewhere, Chris Bertram links to a piece by Charles Ferguson in the Chronicle of Higher Education.

Friday, October 01, 2010

Did economists predict the possibility of the global economic crisis before it occurred? Did they describe sources of instability as they were building up? I think the following three papers are good for exploring these questions:

The answer I get from these articles is mainly negative for orthodox economics. Robert Shiller receives praise. He could be said to be a mainstream economist. But, as I understand it, his analysis was based on behavioral economics and the rejection of the Efficient Market Hypothesis. The empirical evidence suggests macroeconomists should expand research following Wynne Godley's stock-flow consistent models. Imperia and Maffeo point to those who argued that financial innovation was leading to increased debt, an attempt to compensate for reduced aggregate demand resulting from increased income inequality.

Saturday, September 25, 2010

Michael Hirsh recently ("Our Best Minds Are Failing Us", Newsweek, September 16, 2010) lamented how unwilling mainstream economists are to change their thinking in light of the events of the last few years. Brian Milner, a columnist for The Globe and Mail has added to the chorus of complaints.

Perhaps an opportunity now exists for a new introductory textbook in economics that differs fairly comprehensively from mainstream textbooks. Never mind Colander’s approach of modifying his textbooks at most 15% from the previous editions so that mainstream economists will not reject it. Maybe some enterprising heterdox economist should write an uncompromising introduction to economics that is also up-to-date on current events. Years ago, I listed some textbooks. More textbooks have become available since then, for example, G. C. Harcourt’s The Structure of Post-Keynesian Economics: The Core Contributions of the Pioneers. I am not sure that Luigi L. Pasinetti’s Keynes and the Cambridge Keynesians: A ‘Revolution in Economics’ to be Accomplished counts as a textbook. I’m sure I’m leaving much out.

But I’m not sure the packaging on most of these is what I’d like to see tried. The textbook I have in mind should be fairly thick, have various boxed asides, and have problem sets after every chapter. (The problem sets could include essays questions and have less numerical examples than is common.)

1) Heat vegtable or olive oil over medium-high heat in a wide, deep pot (e.g., a Dutch oven). Cut sausage in half lengthwise, then in half lengthwise again. Dice into small pieces and add to pot.

2) Peel and dice onion, adding it to the pot as you do. Dice celery (including leaves) and pepper, adding them to the pot. (Based on what I've seen, one could also add a diced carrot.)

3) When all vegtables are added, cook about 5 minutes so they soften a little. Add bay leaf, thyme, tumeric if desired chicken (e.g., whole thighs) and rice. Add broth plus water to equal 4 cups. Season with a little salt and pepper.

4) Bring to a boil, cover, reduce heat to simmer and cook 30 or 40 minutes, or until chicken is cooked and rice is tender. (I'm thinking of trying it with frozen peas added with about 20 minutes left.)

Sunday, September 12, 2010

1.0 IntroductionCosma Shalizi says, "It is not true that ergodicity is incompatible with sensitive dependence on initial conditions." This poses some questions for me: Can I give an example of a non-ergodic process that also exhibits sensitive sensitive dependence on initial conditions? Can I give an example of an ergodic process that exhibits sensitive dependence on initail conditions?

This post answers the former question. I consider Newton's method for finding roots of unity in the complex plane. The latter question is probably more important for Cosma's assertion. For now, I cite the Lorenz equations as an example of an ergodic process with the desired sensitive dependence.

Cosma's assertion, "It is not true that non-stationarity is a sufficient condition for non-ergodicity," directly contradicts Paul Davidson. I do not address that contradiction here.

2.0 Newton's MethodNewton's method is an algorithm for finding the zeros of a function. In this post, I illustrate the method with the function:

F(z) = z3 - 1,

where z is a complex number. A complex number can be written as a two-element vector:

z = [x, y]T = x + jy

where j is the square root of negative one. (I've been hanging around electrical engineers.) Likewise, one can consider the function F as a vector of two elements:

F(z) = [f1(z), f2(z)]T

The first component maps the real and imaginary parts of the argument to the real part of the function value:

f1(z) = x3 - 3xy2 - 1

The second component maps to the imaginary component of the function value:

f2(z) = y (3x2 - y2)

Newton's method is for numerically finding a solution to the following equation:

F(z) = 0

In my case, one is searching for the cube roots of unity. The method is an iterative method. An initial guess is refined until successive guesses are close enough together that one is willing to accept that the method has converged. A guess is refined by taking a linear approximation to the function at the guess. That guess is refined by solving for the zero of that linear approximation. The zero is the next iteration.

The derivative of a function, when evaluated at the current iterate, provides the linear approximation. The Jacobian is the two-dimensional equivalent of the derivative. The Jacobian, J, is a matrix with the following elements:

Ji, 1([x, y]T) = dfi([x, y]T)/dx, i = 1, 2.

Ji, 2([x, y]T) = dfi([x, y]T)/dy, i = 1, 2.

Newton's method is specified by the following iterative equation:

zn + 1 = zn - J-1(zn) * F(zn)

3.0 Numeric ExplorationsFigure 1 shows a coloring of the plane based on the application of Newton's method. Each point in the plane can be selected as an initial point. The method is applied, and the point is colored according to which of the three cube roots of unity to which the method converges. Figure 2 shows an enlargement of the region around the indicated point in the northeast of Figure 1. Notice the fractal nature of the regions of convergence.

Figure 1: Fractal Basins of Attraction for Newton's Method

Figure 2: An Enlargement of These Fractal Basins

To exhibit sensitive conditions on initial conditions, I wanted to find nearby points whose trajectory diverges under this dynamical process. Table 1 lists six points selected from Figure 2. They fall into three groups, depending on which root they converge to. I claim that one can find at least three distinct points such that each pair is as close as one wants that each converge to a seperate root.

Table 1: Limit Points for Newton's Method

Initial Guess

Limit Point

0.3899 + j 0.6871

1

0.3938 + j 0.6780

(-1/2) - j (31/2)/2

0.3986 + j 0.6811

(-1/2) + j (31/2)/2

0.4010 + j 0.6868

1

0.3980 + j 0.6908

(-1/2) - j (31/2)/2

0.3943 + j0.6949

(-1/2) + j (31/2)/2

Figure 3 and 4 display the trajectories of the six points selected for Table 1. Apparently the function is very shallow in this region. I had not realized before these explorations that these sorts of trajectories go so far from the origin before returning to converge to a root on the unit circle.

Figure 3: Real Part of Some Time Series From Newton's Method

Figure 4: Imaginary Part of Some Time Series From Newton's Method

4.0 ConclusionsCosma provides this definition, among others, of an ergodic process:

"A ... process is ergodic when ... (almost) all trajectories generated by an ergodic process belong to a single invariant set, and they all wander from every part of that set to every other part..."

This definition is appropriate for both deterministic and stochastic processes.

The three roots of unity constitute the non-wandering (invariant) set for the dynamical system created by the above application of Newton's method. A trajectory that has converged to one of the roots does not wander to any other root. So the process is non-ergodic. Yet which root a process converges to is crucially dependent on the initial conditions. A small variation in the initial conditions leads to a long-term divergence in trajectories. This is especially evident because of the fractal structure of the basins of attraction of the three roots.

I think of the above as close to recreational mathematics. I have not tied the above example into any economics model. Common neoclassical models, such as the Arrow-Debreu model of general equilibrium, fail to tie dynamics down. I find it difficult to see how one who has absorbed this fact and understands the mathematics of dynamical systems can find credible much orthodox teaching in economics.

Sunday, September 05, 2010

"Two souls, alas, do dwell within this breast. The one is ever parting from the other" -– Goethe

"He [i.e., Dickens] told me that all the good simple people in his novels, Little Nell, even the holy simpletons like Barnaby Rudge [Slater comments parenthetically that this must have been Dostoevsky's description, not Dickens' -- indeed] are what he wanted to have been, and his villains were what he was (or rather, what he found in himself), his cruelty, his attacks of causeless enmity towards those who were helpless and looked to him for comfort, his shrinking from those whom he ought to love, being used up in what he wrote. There were two people in him, he told me: one who feels as he ought to feel and one who feels the opposite. From the one who feels the opposite I make my evil characters, from the one who feels as a man ought to feel I try to live my life. Only two people? I asked." -- Fyodor Dostoevsky

I have previouslydescribed agents that assess an action by ranking outcomes among a number of incommensurable dimensions. By Arrow's impossibility theorem, such an agent in general cannot have a single aggregate ranking of the outcome of actions.

I was able to list all best choices for my simple example. That is, for each menu, I listed best choices, with ties being possible. (By the way, a budget constraint is a menu.) If one wants to generalize this approach, one would need to specify methods for specifying best choices when listing all possible menus by hand becomes impractical. Pairwise voting is not a good idea, since the results depend on the voting order in which pairs are compared. Furthermore, one would not want to specify one such method, but allow for many different possibilities.

Ulrich Krause has done this. He calls the method for choosing out of these rankings of different aspects an agent's "character". As I understand it, he allows for these rankings to change, based on the agents experience. And so he ends up with a formal model of opinion dynamics.

I don't know if or how this relates to Akerlof's identity dynamics, but, I think, that would be an interesting question to explore.

Monday, August 30, 2010

'But over the long run, money is, as we economists like to say, neutral. This means that no matter what the inflation rate is and no matter what the FOMC does, the real return on safe short-term investments averages about 1-2 percent over the long run.'

This, of course, is false. Communities of economists exist who set their theories in historical time and dispute that money is neutral in any run. I prefer to point to Post Keynesians, but Austrian School economists satisfy these criteria also. Furthermore, economists within such schools surpassed mainstream economists in the current historical conjuncture by having pointedout the possibility of the global financial crisis before its occurrence.

I think economists should strive not to tell untruths abouts what economists believe.

Friday, August 27, 2010

1.0 IntroductionApparently, some have been discussing whether the gross increased inequality in the USA is connected with the depressionary conditions we are in. So I thought I would climb on my bicycle and do some arithmetic.

I take it as a stylized fact that an increase in inequality is associated with an increase in the average and marginal propensity to save.

There's something called the Harrod-Domar model of growth. I'm not sure I've ever read Domar. I've certainly read more of Harrod than I have of Domar. So in the sequel, I refer exclusively to Harrod.

Harrod defined three rates of growth: the actual rate, the warranted rate, and the natural rate. Increased inequality can result in the warranted rate exceeding the natural rate. Since the warranted rate is unstable and the actual rate cannot long exceed the natural rate, increased inequality is likely to lead to the actual rate of growth falling below and away from the warranted rate, that is, to depressions.

2.1 The Actual RateAlong a steady state growth path, the ratio, v, of the value of capital to the value of net income is constant:

v = K/Y,

where K is the value of the capital stock, and Y is the value of net income. v is known as the capital-output ratio. Thus:

dY/dt = (1/v) dK/dt

Investment, I, is defined to be the change in the value of capital with time. Hence,

(1/Y) dY/dt = (1/v) (I/Y)

The left-hand-side of of the above equation is, by definition, the rate of growth, g, of the economy. The equality of investment and savings is an accounting definition in a model with no foreign trade and no government. Therefore,

g = (1/v) (S/Y)

Define the savings rate, s:

s = S/Y

Then, a steady state growth ratio is the ratio of the savings rate to the capital-output ratio:

g = s/v

That is, the (actual) rate of growth is the quotient of the savings rate and the capital-output ratio.

2.2 The Warranted RateSuppose the savings rate and the capital-output ratio are as desired by income recipients (consumers) and firms, respectively. This defines Harrod's warranted rate of growth:

gw = sd/vd

where the subscripts on the right hand side stand for "desired". The warranted rate of growth is being achieved when expectations are being realized and current actions are not setting up forces to disturb current expectations.

The warranted rate of growth extends Keynes' analysis to the long period. Consider the stability of a warranted growth path. If the actual rate of growth exceeds the warranted rate, capacity will be utilized at a greater rate than firms expected. They will increase investment faster than the warranted rate, and the rate of growth will deviate from the warranted rate even more. Likewise, if the actual rate falls below the warranted rate, firms will cut back on investment since the plans upon which their investment was made are not being realized. Hence, the warranted rate is unstable.

Harrod suggested that this instability of the warranted rate is more like an inverted flat-bottomed bowl than a knife-edge.

2.3 The Natural RateSuppose the labor force is initially fully employed. Let n be the rate of growth of the labor force:

n = (dL/dt)/L

Define the value of output produced per employed worker:

f = Y/L

Harrod-neutral technical change occurs when the value of output per worker grows at a constant rate, m, while the rate of profit stays unchanged:

m = (1/f) df/dt

Harrod-neutral technical progress implies that the productivity of labor is growing at the same rate in all industries.

Anyways, the following equation follows:

dY/dt = f dL/dt + L df/dt

Some algebra yields:

(1/Y) (dY/dt) = ( 1/L) (dL/dt) + (1/f) (df/dt)

The left hand side of the above equation is the rate of growth that keeps the labor force fully employed (or a constant percentage unemployed). Harrod calls this the natural rate of growth. Hence, assuming Harrod-neutral technological progress, the natural rate of growth is the sum of the rate of growth of the labor force and the rate of growth of labor productivity.

gn = n + m

3.0 ConclusionsNotice that the determinants of the warranted rate of growth - the savings rate and the desired capital-output ratio - are taken as exogeneous constants. The determinants of the natural rate of growth - the growth of the labor force and Harrod-neutral technological progress - are also given. Hence, the warranted and natural rates can only be equal by a fluke.

Solow, following up on some work by Pivlin, suggested that the desired equality between the warranted and natural rates can be brought about by considering the capital-output ratio as a well-behaved function of the rate of interest. Divergences between the two rates can be corrected by variations in the distribution of income. This approach of neoclassical macroeconomics is exemplified in Solow's eponymous growth model, but it has been shown to be not well-founded in the Cambridge Capital Controversy.

If the warranted rate is below the natural rate, a moderate increase in the saving rate is desirable if the economy is exhibiting boom-like conditions. This would bring the warranted rate towards the actual rate of growth while still keeping it below the natural rate of growth.

Notice that when the warranted rate exceeds the natural rate, the economy must sometime fall below the warranted rate. The natural rate sets a limit which the economy cannot long exceed. Because of the instability of the warranted rate, such an economy will experience frequent and perhaps prolonged recessionary conditions. Since increased savings intensify the discrepancies between the warranted and natural growth rates under these conditions, increased savings intensify the frequency and severity of recessions. That is, increased inequality can intensify the frequency and severity of recessions.

Tuesday, August 24, 2010

... seems often to me to be the main point of many mainstream economists these days. I deliberately don't write, "Why you should listen..." Somebody as stupid as Kartik Athreya, a PhD. with the research department of the Federal Reserve Bank of Richmond, appears to be doesn't deal in arguments. I also see this sort of babble in recent posts by FrancesWoolley, and Mike Moffat. (See also Nick Rowe's comments to those posts.)

Sunday, August 22, 2010

JeffreyMiron teaches EC1017 at Harvard. "A Libertarian Perspective on Economic and Social Policy" is the course title, and PDFs for the lectures are available for download.

Based on the notes for the three lectures I looked at, Miron supposedly derives propertarian policy from intermediate principles (e.g., "efficiency"), with little to no data on relative magnitudes. I don't care for this approach myself, never mind the policy conclusions. He seems to mention no names. The reading list (from Spring 2009) does not include his book (which I haven't read). Perhaps Miron's experience is that Harvard students can be counted on to bring up Rawls, Karl Popper's piecemeal social engineering, Alan Haworth, and even Nozick.

Thursday, August 19, 2010

I had associated the title of this post with the 17th century and the period of the English Civil War. I think it occurs somewhere in Christoper Hill's The World Turned Upside Down: Radical Ideas During the English Revolution. Hill's book is an account of Anabaptists, Diggers, Levellers, Muggletonians, the New Model Army, Ranters, and Quakers - a very heady and confusing mix.

So I was startled yesterday to read the phrase in Crispin: The Cross of Lead. This is a Newberry-prize winning children's book, by Avi. It is set in England in the 14th century. I think it conveys a good idea of the drudgery and isolation of village life at the time; the seemingly unchangable hierarchy; and the bustle, confusion, and filth of a city before modern plumbing. I also like that Christianity is presented as a form of life, a language that all we see cannot but help using.

So is Avi's use of the phrase an anachronism? Hill may reference it, but, if so, the people of his time were harking back to a previous one. Apparently, the phrase is associated with John Ball, the leader of the 1381 Peasants’ Revolt. I know nothing about the Peasants’ Revolt, although Hill does refer to it in one line. But John Ball does appear in Avi’s book. Crispin, our thirteen-year old hero, overhears him conspiring. John Ball says:

"...that no man, or woman either, shall be enslaved, but stand free and equal to one another. That all fees, obligations, and manorial rights be abolished immediately. That land must be given freely to all with a rent of no more than four pennies per acre per year. Unfair taxes must be abolished. Instead of petty tyrants, all laws shall be made by consent of a general commons of all true and righteous men.

Above all persons, our lawful king shall truly reign, but no privileged or corrupt parliaments or councilors.

The church, as it exists, should be allowed to wither. Corrupt priests and bishops must be expelled from our churches.. In their place will stand true and holy priests who shall have no wealth or rights above the common man..."

Update: I've learned a new vocabulary word: A Jacquerie is a peasants' revolt, named after the French peasants' revolt of 1358.

Friday, August 13, 2010

This is mathematics, not economics. It is meant to be an introduction to how abstract mathematicians can be.

2.0 Some Definitions for Set Theory

Two sets are the same size if and only if they can be put into a one-to-one correspondence with each other.

A set S1 is bigger than the set S2 if and only if:

A subset of S1 can be put into one-to-one correspondence with S2, and

S2 cannot be put into one-to-one correspondence with S1.

A set is countably infinite if and only if it can be put into one-to-one correspondence with the set of natural numbers N = {0, 1, 2, ...}. (The integers and the rational numbers are both countably infinite.)

The power set P(S) formed from the set S is the set of all subsets of S. For example, the power set for the set {a, b} contains four elements:

P( {a,b} ) = {S | S ⊂ {a, b}.} = { ∅, {a}, {b}, {a, b} }

3.0 A Theorem

Theorem For all sets S, the power set P(S) is bigger than the set S.

Proof: First, show that a subset of P(S) can be put into one-to-one correspondence with S. Consider the set of singletons:

{ {a} | a is an element of S }.

Since each singleton {a} is a subset of S, the set of all singletons is a subset of P(S). And the set of all singletons maps one-to-one to S.

Next, show, by a proof by contradiction, that P(S) cannot be put into one-to-one correspondence with S. Suppose that there exists a one-to-one function f that maps S into P(S).

Notice that, for all a ∈ S, f(a) is a subset of S. For any given a in S, either

a ∈ f(a)

or

a ∉ f(a).

Define the set T to be the set of all elements in S that map under f to a set not containing themselves:

T = { a | a ∈ S and a ∉ f(a)}

Since f is one-to-one and T is a (possibly empty) subset of S, there exists, by hypothesis, an element b in S such that

f(b) = T.

Now consider whether or not

b ∈ T.

Suppose true. But, by the definition of T as the set of elements of S that are not elements of the subset of S that they map to, b cannot be in f(b), that is, T. But, if b is not in f(b), by the definition of T, b must be in T. So either way yields a contradiction. Thus, no such b can exist.

So I have shown that there does not exist an element b in S that maps under f to T. Yet T is in P(S). Thus, f cannot be one-to-one. Which was to be demonstrated.

4.0 Applying the Theorem to the Set of Natural Numbers

An interesting property of the above proof is that it applies to both finite sets and infinite sets. So start with N, the set of natural numbers. N contains an infinite number of elements. But, by the theorem, P(N), the set of all subsets of the natural numbers, is a set containing a bigger infinity. One can go on to form a set of infinite sets, each with a bigger size infinity:

U0 = { N, P(N), P(P(N)), ..., Pn(N), ...}

(Under the Zermelo Frankel axioms for set theory, the elements of a set do not need to all be of the same "type".) One can repeat the process of forming a sequence of power sets:

U1 = { U0, P(U0), P(P(U0)), ..., Pn(U0), ...}.

One can even imagine constructing a power set of all these difference size infinite sets in this sequence of sequences of sets:

P( { U0, U1, U2), ...} )

The definitions of infinite sets need not stop here.

4.0 ConclusionI don't find the above hard to follow if I think of it as merely a matter of syntactic manipulation of symbols. Do I have a clear idea of these infinities of different size infinities after every point in this sequence of definitions? Does anybody? This is not so clear to me.